# nLab complex volume

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Differential cohomology

differential cohomology

# Contents

## Idea

The Cheeger-Simons classes are complexified secondary invariants.

Under identifying the fundamental class of a hyperbolic 3-manifold $X$ as an element in the Bloch group, the corresponding degree-3 Cheegers-Simons invariant is the complex volume of the 3-manifold, namely the linear combination

$CS + i vol$

of its Chern-Simons invariant and its volume (e.g. Neumann 11, section 2.3, Garoufalidis-Thurston-Zickert 11).

This appears as the action in analytically continued Chern-Simons theory.

(It is, incidentally, also the contribution of a corresponding membrane instanton wrapping a hyperbolic 3-cycle.)

## Properties

### Volume conjecture

The volume conjecture for the Reshetikhin-Turaev construction states that in the classical limit it converges to the complex volume (MMOTY 02, Conjedtcure 1.2, see also Chen-Yang 15)

## References

• Jeff Cheeger, Jim Simons, Differential characters and geometric invariants, in Geometry and Topology, Proceedings of the Special Year, University of Maryland 1983-84, eds. J. Alexander and J. Harer, Lecture Notes in Math. 1167, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 50–80.

• Johan Dupont, Richard Hain, Steven Zucker, Regulators and characteristic classes of flat bundles (arXiv:alg-geom/9202023)

• Walter Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413-474 (arXiv:math/0307092)

• Walter Neumann, Realizing arithmetic invariants of hyperbolic 3-manifolds, Contemporary Math 541, Amer. Math. Soc. 2011, 233–246 (arXiv:1108.0062)

• Stavros Garoufalidis, Dylan Thurston, Christian Zickert, The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds (arXiv:1111.2828, Euclid)

The volume conjecture (Kashaev’s conjecture) for complex volume is due to

• Varghese Mathai, section 6 of $L^2$-analytic torsion, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386